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A Cable Around the World

© Copyright 2001, Jim Loy

This is a famous puzzle. Pretend that the earth is a perfectly smooth sphere with a radius of 6,378 km (3959 mi). And the people on this earth have made a steel cable that goes all the way around the earth at the equator. They meant to make it fit perfectly, resting on the equator, all the way around. But they accidentally made it 1 meter (over 3 ft) too long. They decided to support the cable, so that it is exactly the same distance from the earth, all the way around. How high above the surface of the earth did they place the cable?

Answer:

People are often surprised to hear the answer. Their guesses are usually much too small. The answer is that the cable is 1/(2pi) meters or about 0.159 meters above the surface of the earth. Just use the C=2 pi r formula for the cable, with and without the extra 1 meter (see (pi)). It turns out that it does not matter what the radius of the earth is. The cable could have been made around a basketball. Adding 1 meter to its length adds 1/(2pi) meters to its distance over the sphere. The units do not matter either. If we add one mile to the cable, the distance above the surface would increase by 1/(2pi) miles.

A friend of mine once had the idea that it was inefficient for airliners to travel at great height. Surely, that would add a great distance to their flight paths, because of the curvature of the earth. This is the above problem in reverse. Adding 1 km to the height of an around the world flight only adds 2pi (about 6.28) km to the flight around the world.

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